An abelian category is equivalent to a subcategory of the category of abelian groups

Question

I recently heard the statement that any abelian category is equivalent to a subcategory of Ab, the category of abelian groups.

Is this statement true, and if so where can I find a proof?

Answer 1

Freyd-Mitchell's embedding theorem states that any small abelian category is equivalent to a full subcategory of the category of $R$-modules for some (not necessarily commutative) ring $R$.

If you remove the adjective "full", you can have any small abelian category be equivalent to a subcategory of $\mathbf{Ab}$ - it can be interesting to see how to derive that from the above (be careful though, that $R-\mathbf{mod}$ is not iself a subcategory of $\mathbf{Ab}$). But the notion of "subcategory", when it's not full is... well it's a bit odd, to say the least.

hunter pointed out in their answer that this is not a "serious" obstruction as you can perform most arguments in a small subcategory. This is true to some extent but one has to be careful though, to see which arguments are "local", in the sense that they indeed only depend on a small abelian subcategory containing the objects of interest; and which aren't.

For instance, arguments concerning infinite (co)limits can turn out to fail : it's easy to prove that in $R-\mathbf{mod}$, filtered colimits are exact; but this fails badly in general abelian categories (think about $^{op}$), and this is because "this diagram is a colimit diagram" is not a "local" condition : it depends on the whole category.

So Freyd-Mitchell's embedding theorem can help in understanding "local" statements about abelian categories, but not all statements though, so smallness is not only a useless set-theoretic condition.

For a proof, you can see references here (see the section References)

EDIT : as pointed out in the comments below, if you're not afraid of Grotbendieck universes, then this is not a serious obstruction, though the problems I mentioned above are still there, because exact fully faithful functors need not preserve (co)limits, or injectives/projectives. This seems to actually be the real reason why these types of arguments fail.

Answer 2

This is Freyd's embedding theorem. It's false as stated for set-theoretic reasons; you need the additional hypothesis that the category is small. My understanding is that this is not a serious obstruction, even though most abelian categories you care about are not small, because you can always make whatever argument you wanted to make in a small subcategory; don't know the details though.